Effect of Capillary and Marangoni Forces on Transport Phenomena in

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EFFECT OF CAPILLARY AND MARANGONI FORCES ON TRANSPORT PHENOMENA IN MICROGRAVITY akshay kundan, Joel Plawsky, and Peter C. Wayner Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.5b00428 • Publication Date (Web): 15 Apr 2015 Downloaded from http://pubs.acs.org on May 3, 2015

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EFFECT OF CAPILLARY AND MARANGONI FORCES ON TRANSPORT PHENOMENA IN MICROGRAVITY

Akshay Kundan The Howard P. Isermann Department of Chemical and Biological Engineering Rensselaer Polytechnic Institute Troy, NY, 12180. [email protected] Joel L. Plawsky The Howard P. Isermann Department of Chemical and Biological Engineering Rensselaer Polytechnic Institute Troy, NY, 12180. [email protected] Peter C. Wayner, Jr. The Howard P. Isermann Department of Chemical and Biological Engineering Rensselaer Polytechnic Institute Troy, NY, 12180. [email protected]

ABSTRACT The Constrained Vapor Bubble (CVB) experiment concerns a transparent, simple, "wickless" heat pipe operated in the microgravity environment of the International Space Station (ISS). In a microgravity environment, the relative effect of Marangoni flow is amplified because of highly reduced buoyancy driven flows as demonstrated herein. In this paper, experimental results obtained using a transparent 30 mm long CVB module, 3 x 3 mm in square cross-section, with power inputs of up to 3.125 W are presented and discussed.

Due to the extremely low Bond number and the dielectric materials of

construction, the CVB system was ideally suited to determining if dry-out as a result of Marangoni forces might contribute to limiting heat pipe performance and exactly how that limitation occurs. Using a combination of visual observations and thermal measurements, we find a more complicated phenomenon in which opposing Marangoni and capillary forces lead to flooding of the device. *Corresponding Author: [email protected] 1 ACS Paragon Plus Environment

A simple one-

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dimensional, thermal-fluid flow model describes the essence of the relative importance of the two stresses. Moreover, even though the heater end of the device is flooded and the liquid is highly superheated, boiling does not occur due to high evaporation rates.

KEY WORDS: Marangoni flow, Capillary Pressure Gradient, Heat pipe, Performance Limitation, Dry-out.

1. INTRODUCTION Heat pipes, are passive heat transfer devices frequently used in high heat flux applications. They are especially attractive in microgravity environments where the low Bond number allows for significant heat transfer, and the device's robustness and reliability are extremely attractive features. Heat pipes operate primarily via capillary action. Liquid is evaporated from an extended meniscus at the hot end, flows to the cold end where it condenses in another extended meniscus, and then the cooled liquid flows back towards the hot end via capillary action. Liquid return to the hot end often occurs via a wick, but wickless designs with sharp corners also work well, especially in the case of micro heat pipes. Heat pipes, and especially micro and miniature heat pipes that are essentially unaffected by gravity, are well-understood devices and the equations governing their operation and limitations have achieved much attention [1]. The isothermal, vapor-liquid distribution inside a confined geometry such as a heat pipe was theoretically described by Ajaev and Homsy [2, 3]. Many papers have been published that model the transport performance of heat pipes and especially micro heat pipes, since the latter are unaffected by gravity. Cotter [4] published the original micro heat pipe paper in 1984. Peterson [5] and Faghri [6, 7] have written excellent reviews of micro heat pipe research and development that discuss both experimental investigations and the models developed to predict heat pipe performance. Khrustalev and Faghri [8] and Swanson and Peterson [9] have used models based on the augmented Young-Laplace equation to analyze evaporation processes inside a micro heat pipe. Babin et al. [10] presented a steadystate model that assumes prior knowledge of the liquid film profile in order to compute the liquid and

2

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vapor pressure drops inside the device. Longtin et al. [11] have also developed a one- dimensional model to predict the operating parameters of the micro heat pipe.

Bowman and Maynes [12] developed

analytical expressions for the heat-pipe fin temperature distributions and compared those predictions with a solid fin. Bowman et al. [13, 14] used a lumped inside heat transfer coefﬁcient to describe the thermal performance of a heat pipe with a vapor core and compare that to a solid ﬁn. Heat pipes suffer from a number of performance limitations intimately tied to the dynamics of the evaporating meniscus and the geometry of the device [6]. These limitations include the capillary, boiling, sonic, and entrainment limits. Suman et al. [15] created a model for a “V” grooved, regular polygonal heat pipe geometry and used that model to determine the capillary limit and dry-out length of the device. Xu and Carey [16] developed an analytical model that was used to predict the heat transfer characteristics and the role of disjoining pressure in film evaporation on a microgrooved surface. Stephan and Busse [17] developed a numerical model exploring the details of this extended, evaporating meniscus. Ma and Peterson [18] developed a model to predict the maximum capillary heat transport capacity of a liquid flowing in V-shaped grooves and experimentally verified their predictions. Catton and Stroes [19] developed a semi-analytical model to predict the capillarity limit of a similar, heated inclined triangular groove. Kihm and Pratt [20] demonstrated that interfacial, thermocapillary stresses can be counteracted by introducing naturally occurring concentration gradients associated with distillation in binary fluid mixtures. Migliaccio et al. [21] studied the evaporating meniscus in a V groove and established the importance of the evaporating thin film region to the total meniscus heat transfer. Holm et al. [22] studied an analytical model of the evaporation of liquid in capillary grooves predicting the surface heat transfer coefficients of the wetted surface. Panchamgam et al. [23] have done a comprehensive study of fluid flow and heat transfer occurring in the contact line region of an evaporating meniscus in a miniature heat exchanger. Scriven and Sternling [24] reviewed the details of the long history of the Marangoni effects in 1960. For example, he pointed out that the name was due to the Italian physicist Carlo Marangoni of Pavia and Florence (1840-1925) but the first correct explanation of the spreading of a drop of alcohol on the surface 3


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of water was presented by James Thompson in 1855 [25]. Since then, researchers have studied the role of Marangoni flows in crystal growth [26], alveolar stability of lungs [27], bi-component droplet gasification [28], spreading of liquid mixtures on a solid surface [29] etc. Marangoni flow is also responsible for interfacial turbulence phenomenon observed between two unequilibriated fluids [30], rates of heat and mass transfer [31] and diffusion accompanied by a chemical reaction [32]. It is well known that modified flow patterns associated with Marangoni forces are generated when a significant temperature gradient is present [33 - 35]. Studies on a flat surface have demonstrated that this surface tension gradient is able to induce wetting fluids to climb against the action of gravity. Other Marangoni driven behaviors have also been observed, including fingering instabilities and tear drop formation [35 - 39]. Carles and Cazabat [34] explained the origin of the characteristic flat film associated with a Marangoni-driven coating flow. Boiling within an evaporating meniscus induced by Marangoni flow [40, 41] and the use of Marangoni forces to enhance boiling heat transfer [42] have also been studied. Mathematical models have been developed studying the effect of interfacial phenomena for flow and bubbles in a square channel [43]. The possibility of significant Marangoni flow within the confines of a heat pipe or groove has been discussed and modeled by Yang and Homsy [44], Markos, Ajaev, and Homsy [45], and Savino and Paterna [46]. All these authors have shown that if the interfacial tension decreases with increasing temperature, the resulting Marangoni forces lead to an extended dry-out region at the heater end and a reduction in heat pipe performance. At certain temperatures in water/alcohol solutions, the surface tension of the fluid increases with temperature contrary to the behavior observed in other fluids [47] and is thus termed as “reverse’’ Marangoni effect. Experiments have been carried out on the thermocapillary motion of drops of paraffin oil in an ethanol/water mixture in the reduced gravity environment of the European space sounding rockets [48]. The dry-out predictions from the model [4446] also led to the experimental work of Savino et al [49], di Francescantionio et al [50] and Armijo and Carey [51] who have used water/alcohol mixtures to assist fluid movement to the heater end of the device. Verification of these Marangoni effects have relied primarily on temperature measurements to indicate that "dry-out" was occurring. Ha and Peterson [52] directly observed dry-out by measuring the length of 4


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their evaporator region but their system decoupled evaporation from condensation and eliminated communication between one end of the device and the other. No one has actually observed the liquidvapor distribution or fully modeled that distribution in a complete working heat pipe where significant Marangoni forces may be present. Thus, we do not know whether the physical mechanisms outlined in the models and the heat pipe limitations associated with those models truly occur. In this paper we visually show and thermally identify the effect that the competition between capillary pressure gradient and Marangoni stresses at the hot end of the device on the performance of a "wickless" heat pipe.

The simple, one-dimensional, thermal fluid model gives the essence of the

following: 1) the temperature signature associated with this particular performance limitation; 2) some of the interfacial characteristics of the complicated flow fields; and 3) how the performance limitation arises. In our experiments, we find that very complicated flow fields lead to flooding of the heated end, and not dry-out. Since the flow fields are fully three-dimensional, the complete details of the basic phenomena require extensive future numerical and theoretical analysis.

Figure 1: Conceptual view of the Constrained Vapor Bubble (CVB) experimental apparatus. Not to scale: outside dimensions of 5.5 x 5.5 x 30 mm [53-57]

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2. EXPERIMENTAL MEASUREMENTS The experiments discussed herein were conducted on the International Space Station as part of the Constrained Vapor Bubble program. The heart of the system is shown in Figure 1 and has been discussed in detail in previous publications [53 - 57]. The heat pipe portion of the apparatus was fabricated from a fused silica cuvette. Pentane served as the working fluid. An electrical resistance heater was attached to one end and a thermoelectric cooler was attached to each face of the opposite end to control the overall temperature difference. Most of the heat input to the device was dissipated to the surroundings by radiation. This was unavoidable due to the requirements for visualization and thermal symmetry. The inside dimensions of the device were 3 x 3 x 30 mm. The heat pipe was packaged in a liquid cooled container that allowed for full field of view via a surveillance camera or detailed film thickness measurement using a microscope. Thermocouples were installed at 1.5 mm intervals along the wall of cuvette as well as in the heater and cooler sections. A pressure transducer provided for an overall measurement of the pressure, which demonstrated that the vapor did not contain non-condensables. Voltage and current measurements were used to determine the power supplied to the heater. In this paper, experiments using a 30 mm CVB with electrical power inputs between 0 and 3.125 W are discussed. The Bond number is a dimensionless number measuring the ratio of gravitational force to surface tension force and is given by equation [1]:

=

(1)

ϒ

where, ρ is the density of the fluid, g is the acceleration due to gravity, ϒ is the surface tension of the liquid and h is the appropriate linear dimension. The measured acceleration transient or g-jitter averaged over the period in which experiment was run was found to be 0.19 µg. The Bond number on Earth ranged between 0.8 and 27 whereas on the ISS, based on the g-jitter was between 1.5×10-7 and 5×10-6 over the temperature range of our experiments, 273 K to 463 K.

A. Image data:

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Relatively low magnification surveillance images of the 30mm CVB for different power inputs in microgravity are presented in Figure 2. The vapor-liquid distribution is quite complex, especially at the higher power inputs. At 0W, the constant volume, pure vapor bubble prefers to be attached to the Inconel cold-finger, visible as the dark rectangle on the right. As soon as power is applied to the heater, the bubble migrates toward the heater end. Up to 0.6 W, the temperature gradient is not significant enough to offset capillary flow as can be seen at higher power inputs. The focus of this paper lies within the region defined as the first 10 - 12 mm from the heater end. It is here where the most significant interfacial effects are observed. The extent of this region begins at the heater wall and ends just beyond the bottom of the drop-like region that appears as a light spot near the heater end and is most evident at the higher heat inputs in the higher magnification images presented later in Figure 4. Even in the low magnification images, capillary flow towards and flooding at the hot end are obvious. The thick liquid regions appear as continuous dark areas confined to the corners of the cuvette. understanding of the phenomenon discussed in this paper.

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The images are essential to the

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Figure 2: Low magnification, surveillance images for the 30 mm CVB modules in microgravity as a function of increasing power input. The central vapor bubble is outlined by a dark region representing the thick liquid film in the corners of the cuvette.

The surveillance images of Figure 2 provide only a two-dimensional view of the thickness profile.

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Based on those images, a three-dimensional representation was developed and is shown in the supplemental information as Figure S1. This sketch gives a better view of the morphology of the film in the corners at the heated end and the curvature gradients that exist in those corners.

B. Temperature and Pressure Measurements: The temperature profile along the device was measured using thermocouples installed within the outer wall of the cuvette. Wells (0.45 mm deep) were drilled into the fused silica surface and the thermocouple junction was embedded using thermal paste. The thermocouples were calibrated prior to launch and had a stated accuracy of ±0.5 K. Temperature readings were recorded at 2.40 s intervals once the system reached steady state. The pressure transducer data were collected at the same time intervals as the thermocouple data. The pressure transducer had a range of 37 - 331 kPa with an accuracy of 0.69 kPa and was incorporated primarily to check for system leaks. At room temperature the CVB was at subatmospheric pressure. The temperature data for the 30 mm CVB are presented in Figure 3 for power inputs ranging from 2.2 to 3.125 W. Each curve represents an average of at least twenty separate measurements obtained once the device reached a steady state. These average measurements were then fit using a smoothing spline to generate functions that could be differentiated with as little noise as possible. Figure 3b was generated by differentiating the spline functions fit to Figure 3a.

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(a)

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(b)

Figure 3: (a) Temperature versus position. The green dot(s) represents a line of demarcation between net evaporation of liquid and net condensation of vapor. (b) Temperature gradient versus position for the CVB (pentane as the working fluid) of the 30mm module with various heat inputs in microgravity.

3. RESULTS AND DISCUSSION Using the temperature data presented in Figure 3 for power inputs ranging from 2.2 to 3.125 W. and the surveillance images in Figure 2, we conceptually separated the device into several distinct regions as discussed in [57]. Since we were interested in that part of the heat pipe where we have high temperature gradients, we focused our attention only on the portion of the region to the left of the green dots in Figure 3a. Those dots represent the point where net "evaporation" of liquid changes over to net "condensation" of vapor. We find we can also divide the system into a region above what appears to be a central drop, a region located on the central drop, and a region below the central drop. These regions can be seen in Figure 4a and all exist several millimeters above where condensation begins to occur on the walls of the CVB. The region above the central drop we call the ‘Interfacial flow region’. It is here that the temperature gradient is high enough to develop Marangoni stresses at the vapor-liquid interface that drive liquid pentane in the corner of the cuvette from the heater end toward the cooler end. The Marangoni driven flow faces an opposing flow that occurs due to the capillary pumping from the section below the central drop (Figure 4a and 4b). Below the drop, capillary forces are driving the condensed liquid from the liquid pool at the cooler end toward the heater end. The opposition between Marangoni and capillary flows gives rise to spillover of liquid from the corner of the cuvette onto the flat surface. This excess liquid leads to the formation of the central drop, the pinch point in film thickness at the corner and the thin, curled, junction vortex shown in Figure 4b. The liquid flow from the interfacial flow region and the opposing flow from capillary pumping lead to the formation of counter-rotating vortices inside the central drop. Liquid entering the drop has very little place to go and much of it remains trapped within 10


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the drop waiting to evaporate. As we move across the liquid film from the corner to the liquid-vapor interface, the velocity direction should change from flowing towards the heater near the wall due to the capillary pumping flow to flowing away from the heater near the liquid vapor interface because of Marangoni stress. So, the velocity profile in the corner should resemble a recirculation cell as happens in the coating liquid problems.

(a)

(b)

(c) Figure 4: (a) Expanded view of the interfacial flow region and the forces governing this flow. (b) A sketch of the streamlines in or near the central drop. (c) 10X composite images of the 30 mm µg runs from the heater end to the central drop for varied power inputs.

Looking further at Figure 2 and Figure 4c, one can see that significant interfacial flow likely begins at about 0.6 W of input power as the shape of the film thickness profile in the corner just starts to show a discontinuity in slope. The flow continues to develop beyond this point such that at 1.6 W or higher, the "foot" associated with a Marangoni coating-type flow is apparent. As the power input is steadily increased, the length of the foot grows and the thickness of the foot also increases. As the film 11


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thickness and foot length grow, the conduction resistance through the film increases and so the driving force for evaporating the liquid decreases. Thus, the performance of the heat pipe should decrease with increasing power input once the interfacial region has developed significantly. This scenario is supported by Figure S2 in the supplemental section, where a simple, one dimensional heat transfer model was used to extract information about the internal heat transfer coefficient in the evaporator portion of the 30 mm heat pipe [54, 55]. This coefficient reaches its maximum at about 1.2 W and steadily decreases with increasing power input. Other researchers have observed and modeled a decrease in heat pipe performance as the power level is increased and have associated it with dry-out of the heater end of the device [44-46, 49, 50]. However, in the CVB configuration, we see that dry-out does not occur, but that the origin of the reduction in performance is exactly the opposite phenomenon, flooding of the hot end. Moreover, even though the hot end is flooded, the liquid film thickness is relatively large, and the wall temperature is much higher than the boiling point of pentane at the measured pressure, no boiling is observed up to the limits of operation allowed aboard the ISS. Evaporation from the film surface must still be too high to permit boiling. In addition to the data in Figure S2 whose peak in heat transfer coefficient could be used as an indicator of heater end flooding, we were also interested in whether a simple 1-D heat transfer model could provide more information about the processes occurring in Figure 2 and allow us to locate a temperature profile signature that might indicate the interfacial flooding effect. This would be very useful in cases where visualization is impossible and one is interested in distinguishing between the dry-out and flooding.

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Figure 5: Diagram showing the heat emitted through the CVB by various methods. Pi is the perimeter of the inner surface of the cuvette and Po is the perimeter of the outer surface.

The following simple one-dimensional analytical model, shown schematically in Figure 5 is different from that used in [53-56] to generate Figure S2. Gravity is neglected as is natural convection. Thus, for the glass volume (Ac ∆x), heat conduction through the walls of the device (qcond) is balanced solely by radiative losses from the outer surface (qoutrad) and internal heat exchange due to radiative and phase change (qin) processes. Heat flow rate into the glass volume is positive,

qcond ,x+∆x − qcond ,x = qin ,∆x − qoutrad ,∆x

−kAc

dT dx

+ kAc x+∆x

(2)

dT = qin,∆x − σε Pout ∆x T 4 − T∞4 dx x

(

)

(3)

Here, k is the thermal conductivity of the fused silica, Ac is the cross sectional area of the solid portion of the cuvette, σ is the Stefan-Boltzmann constant and ε is the emissivity of the silica. T∞ is the temperature of the device's enclosure which was held constant by a recirculating cooling fluid through it. Dividing both sides by ∆x and then ∆x → 0, we obtain:

− kAc

d 2T dx

2

=

qin,∆x ∆x

(

− σε Pout T 4 − T∞4

)

(4)

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The heat flow per axial unit length terms are defined as: ' cond

q

≡ − kAc

d 2T

(

' qoutrad = σε Pout T 4 − T∞4

qin' =

(5a)

dx 2

)

(5b)

qin,∆x

(5c)

∆x

Replacing heat derivative terms defined in Eq. 5 to Eq. 4, yields the final form of the overall balance: ' ' qin' = qcond + qoutrad

(6)

Results based on these equations are presented in Figure 6.

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(a)

(b)

(c) Figure 6: (a) Comparison of the external radiative heat loss per unit length for the 30 mm runs (T∞ = 25 o

C) in µg. (b) Comparison of the conduction gradient for 30mm runs in microgravity. (c) Analysis of the

internal rate of heat transfer per unit length for the 30 mm module in µg.

The temperature derivatives and heat flows give us new insight into the prediction of the interfacial flow region in the CVB without looking at images of the vapor-liquid distribution. While

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thermal radiation from the outer surface shows no unusual behavior as shown in Figure 6a, the heat conduction gradient down the glass walls of the CVB shown in Figure 6b and the internal heat transfer rate shown in Figure 6c show a number of characteristic features that indicate the presence of the flooded region. The change in Figure 6c is perhaps the most dramatic. Here one can see that the internal heat transfer rate changes from transferring heat from the wall to transferring heat to the wall near the heater wall. This bending of the curve near the heater end and the presence of a local minimum in the q’in profile is the signature needed to identify heater end flooding formed due to the competition between capillary and Marangoni forces. The local minimum roughly tracks the location of the central drop, especially so at the higher power inputs when the Marangoni stresses are most significant. A one-dimensional fluid flow model based on the liquid film thickness information presented in Figure 7 can be derived to help understand more about the Marangoni flow in the corner and what might be driving it.

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(a)

(b)

(c)

(d)

Figure 7. (a) Measured liquid thickness profile at the wall, hprojected = f (x), in the interfacial flow region. (b) The sketch depicting the apparent film thickness on the wall (hprojected) and its relation with the average film thickness (heff). The experimental thickness observed is the projected liquid profile thickness on the Y axis until the thick region meets the thin film region. (c) The average film thickness obtained at various power inputs. (d) Notations in the interfacial flow region.

In the interfacial flow region, the corner meniscus does not monotonically decrease as one moves from the heater end toward the cooler. The shape resembles a pseudopod or foot characteristic of a 17


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coating flow. The scale of the microscope is 1.281 µ/pixel. So, the wall thickness or hprojected is obtained by multiplying the number of pixels in the gray scale image with the scale of the microscope. The measured thickness profile of the liquid in this region is shown in Figure 7a. The length of the foot and the film thickness are both functions of power input. It is important to note that the thickness measured in Figure 7a is actually a projected meniscus thickness onto the two-dimensional surface imaged. A truer, average film thickness is obtained by considering the general shape of a corner meniscus and how that translates into the gray-scale image we see. The thickness presented in Figure 7b is a good estimate of what we feel the true thickness might be. This approximation leads to the corrected thickness shown in Figure 7c. Since the resistance to the flow of the liquid is much higher toward the thin film region (y axis) than in the axial direction (x axis), and flooding increases with increasing power input, more liquid accumulates in the corner leading to an increased average thickness as shown in Figure 7c. A simple analytical model can be developed that given experimental data on the foot length and average film thickness, allows one to predict the temperature gradient in the liquid-vapor interface of the interfacial flow region as a function of power input as well as the velocity profile in the foot of the interfacial flow region. Looking at a representative fluid element (dx)(dy)(dz) in the middle of the flat film portion of the system and using the following momentum equation to describe the local stress field in a threedimensional liquid film we have:

ρ

∂τ yx ∂τ zx Dv ∂P ∂τ = − L + xx + + + ρg Dt ∂x ∂x ∂y ∂z

(7)

where ρ is the density of the liquid, τ is the shear stress, g is the acceleration due to gravity, PL is the pressure in the liquid, and v is the liquid velocity. The pressure field in the liquid relative to the approximately constant pressure in the vapor, Pv, is a function of the liquid film profile:

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PL = Pv − γκ −

A

(8)

δn

where γ is the interfacial tension, κ is the interface curvature, A is the Hamaker constant, and δ is the local liquid film thickness. Considering the additional effects of phase change, the complete description of the transport processes is extremely complex. However, results obtained using the following very simple model, based on experimental observations, give considerable insight and direction for future experiments. From here on we avoid modeling the extremely thin liquid regions on the flat surfaces away from the corners where the disjoining pressure (last term in Eq. 8) is important. This region is significantly more complicated and awaits future analysis.

Assumptions: In case of microgravity, g =0. Focusing on one-dimensional axial flow in a sub-region, vx(y), with vy =vz =0. For steady state with no acceleration in the x-direction in the flat region, (dvx/dx = 0).

Using these assumptions, Eq. 7 reduces to the following simplified form:

−

∂ PL ∂τ yx + =0 ∂x ∂y

(9)

The shear stress on y surface in x-direction τyx is given by:

τ yx = µ

dvx dy

(10)

where µ is the viscosity of the liquid. Assuming that the pressure stress gradient due to cohesion self-adjusts to a constant value over the distance L, gives:

dPL ∆PL γ = = (κ 2 − κ 1 ) dx L L

(11)

19


Langmuir

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Page 20 of 51

Combining Eqs. 10 and 11 gives:

d 2 vx dy 2

=

γ (κ − κ ) µL 2 1

(12)

The boundary conditions used are: 1. Marangoni surface stress: τ = τh, y =heff

(13a)

2. No slip at the wall (solid-liquid interface): vx =0, y= 0

(13b)

Integrating Eq. 12 using the boundary conditions given in Eqs. 13a and 13b, we arrive at Eq. 14 to determine the velocity field in the coating film region. =

− − ℎ +

(14)

The flux due to phase change in the coating film region is given by Eq. 15. Here, Γ represents a volumetric flow per unit length of liquid evaporating or condensing on the film surface.

= !% #$$ "

(15)

Integrating Eq. 15 using vx from Eq. 14, the coating film thickness, heff, can be obtained and is given by Eq. 14. ℎ =

& ' ( ')

* −

+, #$$

(16)

Near the heated end, κ2 (Figure 7d) is composed of 2 positive radii of curvature meaning both curvatures act to suck liquid from the comparatively cooler end near the central drop whereas near the central drop, κ1 has 2 opposite radii of curvature, one acts to pump the liquid towards the heater end and the other to draw liquid from the heater end and thus, the overall curvature, κ1 is negligible. The shear stress at the liquid-vapor interface is defined by Eq. 17 where T is the temperature.

. =

/

/

=

/ /0

/0 /

(17)

The temperature gradient obtained in Figure 3 is present at the wall of the cuvette. The 20


Page 21 of 51

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Langmuir

temperature gradient at the liquid-vapor interface is unknown. The plot of surface tension as a function of the average liquid temperature in the Marangoni dominated region gives us the average temperature derivative of the surface tension. The slope is constant in Figure 8a, so a constant value of the temperature derivative of surface tension is used. Inserting Eq.17 in Eq. 16 and rearranging the terms, we arrive at Eq. 18 to determine the unknown temperature gradient. /0 /

=

ℎ233 4 5 4) 6

1

78

=6 =>

9

:;

ℎ233

Effect of Capillary and Marangoni Forces on Transport Phenomena in

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